Area Formulas for 2D Shapes

Understanding area formulas for 2D shapes is essential in geometry. These formulas help calculate the space within various figures, from rectangles to circles, and reveal the relationships between dimensions. Mastering these concepts lays the groundwork for more complex geometric studies.

1. Rectangle: A = length × width

   • The area is calculated by multiplying the length and width of the rectangle.
   • Both dimensions must be in the same unit for accurate results.
   • A rectangle can be considered a special case of a parallelogram.

2. Square: A = side²

   • All four sides of a square are equal in length.
   • The area is found by squaring the length of one side.
   • A square is a specific type of rectangle.

3. Triangle: A = ½ × base × height

   • The base can be any side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex.
   • The formula shows that the area is half of the rectangle that could be formed around the triangle.
   • Different types of triangles (e.g., equilateral, isosceles) can use the same formula.

4. Parallelogram: A = base × height

   • The base can be any side, and the height is the perpendicular distance from that base to the opposite side.
   • The area is equivalent to that of a rectangle with the same base and height.
   • Opposite sides of a parallelogram are equal in length.

5. Trapezoid: A = ½(base₁ + base₂) × height

   • The formula averages the lengths of the two parallel sides (bases) and multiplies by the height.
   • The height is the perpendicular distance between the two bases.
   • Trapezoids can be classified as isosceles or right, affecting their properties.

6. Circle: A = πr²

   • The area is determined by squaring the radius and multiplying by π (approximately 3.14).
   • The radius is the distance from the center to any point on the circle.
   • The formula highlights the relationship between the radius and the area of the circle.

7. Regular Polygon: A = ½ × perimeter × apothem

   • The perimeter is the total length of all sides, and the apothem is the perpendicular distance from the center to a side.
   • This formula applies only to regular polygons, where all sides and angles are equal.
   • The area can also be calculated using the number of sides and the length of each side.

8. Rhombus: A = ½ × diagonal₁ × diagonal₂

   • The area is calculated using the lengths of the two diagonals, which intersect at right angles.
   • A rhombus is a type of parallelogram with all sides equal.
   • The diagonals bisect each other and can be used to find the area easily.

9. Ellipse: A = π × semi-major axis × semi-minor axis

   • The semi-major axis is the longest radius, while the semi-minor axis is the shortest radius of the ellipse.
   • The area formula shows the relationship between the two axes and the overall area.
   • Ellipses can represent orbits and other natural phenomena.

10. Sector of a Circle: A = ½ × r² × θ (where θ is in radians)

• The area of a sector is a fraction of the area of the entire circle, based on the angle θ.
• The radius (r) must be in the same unit for accurate area calculation.
• This formula is useful in applications involving circular segments, such as pie charts.